Atoms having low ionization potentials often transfer an electron to atoms having high electron affinities, forming ionic bonds. In an ionically bonded species, such as sodium fluoride (Na⁺F⁻), the atoms are held together by the electrostatic attraction of the opposite charges. In sodium fluoride, both Na⁺ and F⁻ have filled second shells, and each has achieved the stable electronic configuration of the noble gas Ne. In potassium chloride (K⁺Cl⁻), both ions have the electronic configuration of Ar, another noble gas (Fig. 1.3).

Ionically bonded compounds are traditionally the province of inorganic chemistry. Nearly all of the compounds of organic chemistry are bound not by ionic bonds but rather by covalent bonds, which are bonds formed by the sharing of electrons.

1.2a Quantum Numbers We have just developed pictures of some atoms and ions. Let’s elaborate a bit to provide a fuller picture of atomic orbitals. Your reward for bearing with an increase in complexity will be a much-increased ability to think about structure and reactivity. The most useful models for explaining and predicting chemical behavior focus on the qualitative aspects of atomic (and molecular) orbitals, so it is time to learn more about them.

The electrons in atoms do not occupy simple circular orbits. To describe an electron in the vicinity of a nucleus, Erwin Schrödinger (1887–1961) developed a formula called a wave equation. Schrödinger recognized that electrons have properties of both particles and waves. The solutions to Schrödinger’s wave equation, called wave functions and written Ψ (pronounced “sigh”), have many of the properties of waves. They can be positive in one region, negative in another, or zero in between.

An orbital is mathematically described by a wave function, ψ, and the square of the wave function, ψ², is proportional to the probability of finding an electron in a given volume. There are regions or points in space where ψ and ψ² are both 0 (zero probability of finding an electron in these regions), and such regions or points are called nodes. However, ψ² does not vanish at a large distance from the nucleus but rather maintains a finite value, even if inconsequentially small.

Each electron is described by a set of four quantum numbers. Quantum numbers are represented by the symbols n, l, m_l, and s. The first two quantum numbers define the orbital of the electron. The first one, called the principal quantum number, is represented by n and may have the integer values n = 1, 2, 3, 4, and so on. It is related to the distance of the electron from the nucleus and hence to the energy of the electron. It describes the atomic shell the electron occupies. The higher the value of n, the greater the average distance of the electron from the nucleus and the greater the electron’s energy. The principal quantum number of the highest-energy electron of an atom also determines the row occupied by the atom in the periodic table. For example, the electron in H and the two electrons in He are all n = 1, as Table 1.2 shows, and so these two atoms are in the first row of the table. The principal quantum number in Li, Be, B, C, N, O, F, and Ne is n = 2, which tells us that electrons can be in the second shell for these atoms and places the atoms in the second row. The elements Na, Mg, Al, Si, P, S, Cl, and Ar are in the third row, and orbitals for the electrons in these atoms correspond to n = 1, 2, or 3.

 

The second quantum number, l, is related to the shape of the orbital and depends on the value of n. It may have only the integer values l = 0, 1, 2, 3, . . . , (n − 1). So, for an orbital for which n = l, l must be 0; for n = 2, l can be 0 or 1; and for n = 3, the three possible values of l are 0, 1, and 2.

Each value of l signifies a different orbital shape. We shall learn about these shapes in a moment, but for now just remember that each shape is represented by a letter. The orbital for which l = 0 is spherical, and the letter s is used to designate all spherical orbitals. For higher values of l, we do not have the convenience of easily remembered letters the way we do with “s for spherical.” Instead, you just have to remember that p is used for orbitals for which l = 1, d is used for those for which l = 2, and f is used for l = 3. These letters associated with the various values of l lead to the common orbital designations shown in Table 1.3.

 

The third quantum number, m_l, depends on l. It may have the integer values −l, . . . , 0, . . . , +l, and is related to the orientation of the orbital in space. Table 1.4 presents the possible values of n, l, and m_l for n = 1, 2, and 3. Orbitals of the same shell (n) and the same shape (l) are at the same energy regardless of the m_l value.

 

Finally, there is s, the spin quantum number, which may have only the two values ±1/2.

Table 1.5 lists all the possible combinations of quantum numbers through n = 3.

TABLE 1.5 Possible Combinations of Quantum Numbers for n = 1, 2, and 3

 

Be careful. The s that designates the spin quantum number is not the same as the s in a 1s or 2s orbital. What is electron spin anyway? The word spin tries to make an analogy with the macroscopic world—in some ways, the electron behaves like a spinning top that can spin either clockwise or counterclockwise.

As shown in Tables 1.3–1.5, orbitals are designated with a number and a letter—1s, 2s, 2p, and so on. The number in the designation tells us the n value for a given orbital, and the letter tells us the l value. Thus, the notation 1s means the orbital for which n = 1 and l = 0. Because l values can be only 0 to n − 1, 1s is the only orbital possible for n = 1. The 2s orbital has n = 2 and l = 0, but now l can also be 1 (n − 1 = 2 − 1 = 1), and so we also have 2p orbitals, for which n = 2, l = 1. For the 2p orbitals, m_l, which runs from 0 to ±l, may take the values −1, 0, +1. Thus there are three 2p orbitals, one for each value of m_l. These equi-energetic orbitals are differentiated by arbitrarily designating them as 2p_x, 2p_y, or 2p_z. A little later we will see that the x, y, z notation indicates the relative orientation of the 2p orbitals in space.

For n = 3, we have orbitals 3s (n = 3, l = 0) and 3p (n = 3, l = 1). Just as for the 2p orbitals, m_l may now take the values −1, 0, +1. The three 3p orbitals are designated as 3p_x, 3p_y, and 3p_z. With n = 3, l can also be 2 (n − 1 = 2), so we have the 3d (n = 3, l = 2) orbitals, and m_l may now take the values −2, −1, 0, +1, and +2. Thus there are five 3d orbitals. These turn out to have the complicated designations 3d_x²-y², 3d_z², 3d_xy, 3d_yz, and 3d_xz. Mercifully, in organic chemistry we very rarely have to deal with 3d orbitals and need not consider the f orbitals, for which n = 4 and are even more complicated.

For all of the orbitals in Tables 1.4 and 1.5, the spin quantum number s may be either +1/2 or −1/2. We may now designate an electron occupying the lowest-energy orbital, 1s, as either 1s with s = +1/2 or 1s with s = −1/2 but nothing else. Similarly, there are only two possibilities for electrons in the 2p_x orbital: 2p_x with s = +1/2 or 2p_x with s = −1/2. The same is true for all orbitals—3p_z, 4d_xy, or whatever. Only two values are possible for the spin quantum number. That is why it is impossible for more than two electrons to occupy any orbital!

The convention used to designate electron spin shows the electrons as up-pointing (↑) and down-pointing (↓) arrows. Two electrons having opposite spins are denoted ⇅ and are said to have paired spins. Two electrons having the same spin are denoted ⇈ and are said to have parallel spins, or unpaired spins.

How many electrons occupy a given orbital in an atom is indicated with a superscript. When we write 1s² we mean that the 1s orbital is occupied by two electrons, and these electrons must have opposite (paired, with s = +1/2 and s = −1/2) spin quantum numbers. The designation 1s³ is meaningless because there is no way to put a different third electron in any orbital. No two electrons may have the same values of the four quantum numbers. This rule is called the Pauli principle after Wolfgang Pauli (1900–1958), who first articulated it in 1925. These ideas are summarized in Figure 1.4.

 

In Table 1.6, we see the entry 1s ², which means there are two electrons in the 1s orbital. It would seem that 1s¹ would be the appropriate notation for a 1s orbital occupied by one electron. That notation is rarely used, however, and the superscript “1” is almost always understood.

1.2b Electronic Configurations We can now use the quantum numbers n, l, m_l, and s to write electronic descriptions, called configurations, for atoms using what is known as the aufbau principle (aufbau is German for “building up” or “construction”). This principle simply makes the reasonable assumption that we should fill the available orbitals in order of their energies, starting with the lowest-energy orbital. To form these descriptions of neutral atoms, we add electrons until they are equal to the number of protons in the nucleus. Table 1.6 gives the electronic configurations of H, He, Li, Be, and B.

 

For carbon, the atom after boron in the periodic table, we have a choice to make in adding the last electron. The first five electrons are placed as in boron, but where does the sixth electron go? One possibility would be to put it in the same orbital as the fifth electron to produce the electronic configuration ₆C = 1s²2s²2p_x² (Fig. 1.5a). In such an atom, the spins of the two electrons in the 2p_x orbital must be paired (opposite spins).

Alternatively, the sixth electron in the carbon atom could be placed in another 2p orbital to produce ₆C = 1s²2s²2p_x2p_y (Fig. 1.5b). The only difference is the presence of two electrons in a single 2p orbital (₆C = 1s²2s²2p_x²) in Figure 1.5a versus one electron in each of two different 2p orbitals (₆C = 1s²2s²2p_x2p_y) in Figure 1.5b. The three 2p orbitals are of equal energy (because each has n = 2, l = 1), so how should we make this choice? Electron–electron repulsion would seem to make the second arrangement the better one, but there is still another consideration. When two electrons occupy different but equi-energetic orbitals, their spins can either be paired (⇅) or unpaired (⇈). Hund’s rule (Friedrich Hund, 1896–1997) holds that for a given electron configuration, the state with the greatest number of unpaired (parallel) spins has the lowest energy. So the third possibility for carbon’s sixth electron (Fig. 1.5c) is actually the best one in this case. Essentially, giving the two electrons the same spin (⇈) ensures that they cannot occupy the same orbital and tends to keep them apart. So, Hund’s rule tells us that the configuration ₆C = 1s²2s²2p_x2p_y with parallel spins for the two electrons in the 2p orbitals is the lowest-energy state for a carbon atom.

Now we can write electronic configurations for the rest of the atoms in the second row of the periodic table (Table 1.7). Notice that for nitrogen we face the same kind of choice just described for carbon. Does the seventh electron go into an already occupied 2p orbital or into the empty 2p_z orbital? The configuration in which the 2p_x, 2p_y, and 2p_z orbitals are all singly occupied by electrons that have the same spin is lower in energy than any configuration in which a 2p orbital is doubly occupied. Again, Hund rules!

To understand the properties and reactions of the molecules you will encounter in organic chemistry, it will be necessary to know the shapes of the molecules. It turns out that molecular shape can be best understood by knowing where the electrons are. Recall that ψ² is related to the electron density around a nucleus. Therefore, a graph that plots ψ² as a function of r, the distance from the nucleus, can describe the shape of the orbital. Figure 1.6 plots ψ² as a function of r for the lowest-energy solution to the Schrödinger equation, which describes the 1s orbital. The graph shows that the probability of finding an electron falls off sharply in all directions (x, y, and z) as we move out from the nucleus. Because there is no directionality to r, we find the 1s orbital is symmetrical in all directions; that is, it is spherical. As noted earlier, the quantum number l is associated with orbital shape, and all s orbitals, for which l always equals zero, are spherically symmetric. Note also that the electron density never goes to zero—a finite (though very, very small) probability exists of finding an electron at a distance of several angstroms (Å; 1.0 Å = 0.10 nm = 1.0 × 10⁻¹⁰ meters) from the nucleus.

 

⁵There are several units of energy in use. Organic chemists commonly use kilocalories per mole (kcal/mol); physicists use the electron volt (eV). One electron volt/molecule translates into about 23 kcal/mol. Recently, the International Committee on Weights and Measures suggested that the kilojoule (kJ) be substituted for kilocalorie. So far, organic chemists in some countries, including the United States, seem to have resisted this suggestion. We will provide kcal/mol and parenthetical kJ/mol values throughout the text (1 kcal is equal to 4.184 kJ).

⁶Worked Problems are answered in whole or in part in the text. When only part of a problem is worked, that part will have an asterisk. Complete answers can be found in the Study Guide.